TITLE: Duality between Tor and Ext?
QUESTION [5 upvotes]: Let $A$ be a $\mathbb{N}$-graded, locally finite $\Bbbk$-algebra, $\Bbbk$ being a field, $A=\oplus_{n \geq 0} \ A^n$, each $A^i$ being finitely dimensional as a $\Bbbk$ vector space. Assume also that $A$ is connected, i.e. $A^0=\Bbbk$.
Let $A^\ast$ denote its graded dual, i.e. $A^\ast=\oplus_{n \geq 0} \ \mathrm{Hom}_\Bbbk(A^n,\Bbbk)$, which is a connected, graded $\Bbbk$-coalgebra.
Starting from some computations involving the (co)bar complexes for Hochschild (co)homology, I came up with the following question:
What is the connection between $\mathrm{Tor}_\bullet^A(\Bbbk,\Bbbk)$ and $\mathrm{Ext}^\bullet_{A^\ast} (\Bbbk,\Bbbk)$ ?
Computing them using the normalized (co)bar complexes, one obtains, in both cases, bigraded structures, $\mathrm{Ext}^\bullet_{A^\ast} (\Bbbk,\Bbbk)$ being the Yoneda algebra of $A^\ast$ and $\mathrm{Tor}_\bullet^A(\Bbbk,\Bbbk)$ being a coalgebra, obtained in a "dual" way as for the Yoneda algebra.
It would be great to have a duality between these structures, but I don't know how to approach an attempt of proof.
Thank you.

REPLY [8 votes]: If $k$ is a field as you say it is, it is not difficult to see using the standard tensor-hom adjunction that $Ext^*_R(k,k)\cong (Tor_*^R(k,k))^*$ as $k$-graded modules, where $R$ is a $k$-algebra. I've learned this from these Tor-Ext notes by May.
Here's the relevant passage, which contains other comments which might be useful to you. It's at the very end of the note: